More results on congruent modules

W.R. Scott characterized the infinite abelian groups GG for which H≅GH≅G for every subgroup HH of GG of the same cardinality as GG [W.R. Scott, On infinite groups, Pacific J. Math. 5 (1955) 589–598]. In [G. Oman, On infinite modules MM over a Dedekind domain for which N≅MN≅M for every submodule NN of cardinality |M||M|, Rocky Mount. J. Math. 39 (1) (2009) 259–270], the author extends Scott’s result to infinite modules over a Dedekind domain, calling such modules congruent  , and in a subsequent paper [G. Oman, On modules MM for which N≅MN≅M for every submodule NN of size |M||M|, J. Commutative Algebra (in press)] the author obtains results on congruent modules over more general classes of rings. In this paper, we continue our study.